Cutting board sandpaper?
 

Is there something to use to smooth a very old cutting board that has deep grooves in it?

There are times when it is necessary to buy a new one.

It is probably worn and will dull you blades. A new one would be better. They do dry out. I purchased a new one a month ago and could not beleive the difference in my cutting.

Use the other side.

Yes they do dry out, about once a week I lightly spray with water and let it air dry. I also use a mat smoother I saw at Hancock fabrics, it really does make a difference. I use the smoother about once a month. I also clean my mat with alcohol once a month, hence spraying with a very,very fine mist of water and letting it dry. I also clean rulers, table tops, and other stuff with the alcohol, I was amazed how dirty my "clean" rulers were. My mat is over 10 years old and still going strong.
Carol